An isomorphic version of the Busemann–Petty problem for arbitrary measures

Abstract

The Busemann–Petty problem for an arbitrary measure $\mu$ with non-negative even continuous density in ${\mathbb R}^n$ asks whether origin-symmetric convex bodies in ${\mathbb R}^n$ with smaller $(n-1)$-dimensional measure $\mu$ of all central hyperplane sections necessarily have smaller measure $\mu .$ It was shown in Zvavitch (Math Ann 331:867–887, 2005) that the answer to this problem is affirmative for $n\le 4$ and negative for $n\ge 5$. In this paper we prove an isomorphic version of this result. Namely, if $K,M$ are origin-symmetric convex bodies in ${\mathbb R}^n$ such that $\mu (K\cap \xi ^\bot )\le \mu (M\cap \xi ^\bot )$ for every $\xi \in {\mathbb S}^{n-1},$ then $\mu (K)\le \sqrt{n}\ \mu (M).$ Here $\xi ^\bot$ is the central hyperplane perpendicular to $\xi .$ We also study the above question with additional assumptions on the body $K$ and present the complex version of the problem. In the special case where the measure $\mu$ is convex we show that $\sqrt{n}$ can be replaced by $cL_n,$ where $L_n$ is the maximal isotropic constant. Note that, by a recent result of Klartag, $L_n \le O(n^{1/4})$. Finally we prove a slicing inequality

$$\begin{aligned} \mu (K)\le C n^{1/4} \max _{\xi \in {\mathbb S}^{n-1}} \mu (K \cap \xi ^\perp )\ \mathrm{vol}_{n}(K)^{\frac{1}{n}} \end{aligned}$$

for any convex even measure $\mu$ and any symmetric convex body $K$ in ${\mathbb R}^n,$ where $C$ is an absolute constant. This inequality was recently proved in Koldobsky (Adv Math 254:33–40, 2014) for arbitrary measures with continuous density, but with $\sqrt{n}$ in place of $n^{1/4}.$